%I #12 Feb 02 2024 17:49:30

%S 1,1,3,7,18,43,112,282,740,1940,5182,13916,37826,103391,284815,788636,

%T 2195414,6137025,17223354,48495640,136961527,387819558,1100757411,

%U 3130895452,8922294498,25470279123,72823983735,208515456498,597824919725,1716072103910,4931540188084

%N Number of unlabeled loop-graphs covering n vertices such that it is possible to choose a different vertex from each edge (choosable).

%C These are covering loop-graphs with at most one cycle (unicyclic) in each connected component.

%H Andrew Howroyd, <a href="/A369200/b369200.txt">Table of n, a(n) for n = 0..500</a>

%F First differences of A369145.

%F Euler transform of A369289 with A369289(1) = 1. - _Andrew Howroyd_, Feb 02 2024

%e Representatives of the a(1) = 1 through a(4) = 18 loop-graphs (loops shown as singletons):

%e {{1}} {{1,2}} {{1},{2,3}} {{1,2},{3,4}}

%e {{1},{2}} {{1,2},{1,3}} {{1},{2},{3,4}}

%e {{1},{1,2}} {{1},{2},{3}} {{1},{1,2},{3,4}}

%e {{1},{2},{1,3}} {{1},{2,3},{2,4}}

%e {{1},{1,2},{1,3}} {{1},{2},{3},{4}}

%e {{1},{1,2},{2,3}} {{1,2},{1,3},{1,4}}

%e {{1,2},{1,3},{2,3}} {{1,2},{1,3},{2,4}}

%e {{1},{2},{3},{1,4}}

%e {{1},{2},{1,3},{1,4}}

%e {{1},{2},{1,3},{2,4}}

%e {{1},{2},{1,3},{3,4}}

%e {{1},{1,2},{1,3},{1,4}}

%e {{1},{1,2},{1,3},{2,4}}

%e {{1},{1,2},{2,3},{2,4}}

%e {{1},{1,2},{2,3},{3,4}}

%e {{1},{2,3},{2,4},{3,4}}

%e {{1,2},{1,3},{1,4},{2,3}}

%e {{1,2},{1,3},{2,4},{3,4}}

%t brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];

%t Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{1,2}]], Union@@#==Range[n]&&Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]]],{n,0,4}]

%Y Without the choice condition we have A322700, labeled A322661.

%Y Without loops we have A368834, covering case of A134964.

%Y For exactly n edges we have A368984, labeled A333331 (maybe).

%Y The labeled version is A369140, covering case of A368927.

%Y The labeled complement is A369142, covering case of A369141.

%Y This is the covering case of A369145.

%Y The complement is counted by A369147, covering case of A369146.

%Y The complement without loops is A369202, covering case of A140637.

%Y A000085, A100861, A111924 count set partitions into singletons or pairs.

%Y A000666 counts unlabeled loop-graphs, labeled A006125 (shifted left).

%Y A006129 counts covering graphs, unlabeled A002494.

%Y A007716 counts non-isomorphic multiset partitions, connected A007718.

%Y A129271 counts connected choosable simple graphs, unlabeled A005703.

%Y A133686 counts choosable labeled graphs, covering A367869.

%Y Cf. A000088, A000612, A006649, A014068, A055621, A137916, A137917, A368835, A369194, A369199.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jan 23 2024

%E a(7) onwards from _Andrew Howroyd_, Feb 02 2024